feynman’s operational calculi: spectral theory for noncommuting self-adjoint operators

Math Phys Anal Geom (2007) 10:65–80DOI 10.1007/s11040-007-9021-8

Feynman’s Operational Calculi: Spectral Theoryfor Noncommuting Self-adjoint Operators

Brian Jefferies · Gerald W. Johnson ·Lance Nielsen

Received: 3 November 2006 / Accepted: 16 April 2007 /Published online: 29 June 2007© Springer Science + Business Media B.V. 2007

Abstract The spectral theorem for commuting self-adjoint operators alongwith the associated functional (or operational) calculus is among the most use-ful and beautiful results of analysis. It is well known that forming a functionalcalculus for noncommuting self-adjoint operators is far more problematic.The central result of this paper establishes a rich functional calculus for anyfinite number of noncommuting (i.e. not necessarily commuting) bounded,self-adjoint operators A1, . . . , An and associated continuous Borel probabilitymeasures μ1, · · · , μn on [0, 1]. Fix A1, . . . , An. Then each choice of an n-tuple(μ1, . . . , μn) of measures determines one of Feynman’s operational calculiacting on a certain Banach algebra of analytic functions even when A1, . . . , An

are just bounded linear operators on a Banach space. The Hilbert space settingalong with self-adjointness allows us to extend the operational calculi wellbeyond the analytic functions. Using results and ideas drawn largely from theproof of our main theorem, we also establish a family of Trotter product typeformulas suitable for Feynman’s operational calculi.

B. JefferiesSchool of Mathematics, The University of New South Wales,Sydney 2052, Australiae-mail: [email protected]

G. W. JohnsonDepartment of Mathematics, 333 Avery Hall, The University of Nebraska,Lincoln, Lincoln, NE 68588-0130, USAe-mail: [email protected]

L. Nielsen (B)Department of Mathematics, Creighton University,Omaha, NE 68178, USAe-mail: [email protected]

66 B. Jefferies et al.

Keywords Noncommuting self-adjoint operators · Spectral theories ·Feynman’s operational calculi · Disentangling

Mathematics Subject Classifications (2000) Primary 47A13 · 47A60 ·Secondary 46J15

1 Introduction

Let X be a Banach space and suppose that A1, . . . , An are noncommutingelements in L(X), the space of bounded linear operators on X. Further, foreach i ∈ {1, . . . , n}, let μi be a continuous probability measure defined onB([0, 1]), the Borel class of [0, 1]. (Recall that a measure μ is continuousprovided that μ({s}) = 0 for every single point set {s}.) Such measures de-termine an operational calculus or ‘disentangling map’ Tμ1,...,μn from a com-mutative Banach algebra D(A1, . . . , An), called the ‘disentangling algebra’ ofanalytic functions into the noncommutative Banach algebra L(X). (See [4] orDefinition 1.1 below.) It is natural to seek conditions under which suchan operational calculus can be extended beyond the analytic functions inD(A1, . . . , An). Theorem 2.2, the main result of this paper will show, inconjunction with results from [6], that when X = H is a Hilbert space andA1, . . . , An are self-adjoint, the domain of each of the operational calculi ismuch richer than D(A1, . . . , An).

Feynman developed ‘rules’ for his operational calculus for noncommutingoperators while discovering the famous perturbation series and Feynmangraphs of quantum electrodynamics. By the time he wrote [2] he realizedthat this operational calculus could be developed into a widely applicablemathematical technique. Feynman was aware that his work was far from beingmathematically rigorous (see page 108 of [2]), especially with regard to the‘disentangling’ process, the central operation of his functional calculus. Heregarded his operational calculus as a kind of generalized path integral. (SeeSection 14.3 of [10].)

We now give a brief description of Feynman’s heuristic rules:

(a) Attach time indices to the operators to keep track of the order of theoperators in products. Operators with smaller (or earlier) time indicesare to act before operators with larger (or later) time indices no matterhow they are ordered on the page.

(b) With time indices attached, functions of the operators are formed just asif they were commuting.

(c) Finally, the operator expressions are to be restored to their natural order;this is the so-called disentangling process. This final step is often difficult;it consists roughly of manipulating the operator expressions until theirorder on the page is consistent with the time ordering.

How does one accomplish (a)–(c)? There have been several quite variedapproaches to this subject. Many of the references can be found in one of

Feynman’s operational calculi: spectral theory 67

the books [10, 13]. We also call attention to the recent monograph by B.Jefferies [3] and the 1968 paper by Taylor [18]. Both of these are essentiallyconcerned with the Weyl calculus for a finite number of noncommutingbounded operators. The paper [18] focused on operators which are also self-adjoint. The work begun by Maslov [12] and pursued by him and by severalothers is the furthest developed. See especially the book [13] by Nazaikinskii,Shatalov, and Sternin.

We will follow the approach initiated recently by Jefferies and Johnson([4–7]) and further developed by them, Nielsen and others ([8, 9, 11, 14]). Alarge family of operational calculi is defined at one time in this approach. Thisallows us to study a variety of operational calculi within one framework. Italso permits us to solve a wide variety of evolution equations using variousexponential functions of sums of noncommuting operators. (This was carriedout in [8], Section 4, and we hope to pursue this further in later work.) Finally,one can sometimes get information about one (or one type of) operationalcalculus by showing that it is the limit of simpler operational calculi. Indeed,the main theorem of this paper will rest in large part on such an argument.

Johnson and Nielsen established a stability theorem for Feynman’s opera-tional calculi [11] which will supply one of the central facts that we will need forour main result. We state that theorem now even though the precise definitionsof the disentangling algebra and the disentangling map will be postponed untilfurther on in this section.

Theorem 1.1 For each i = 1, . . . , n, let μi and μik, k = 1, 2, . . . be continu-ous probability measures on B([0, 1]) and suppose that the sequence (μik)

converges weakly to μi (denoted μik ⇀ μi) as k → ∞. Then for every f ∈D(A1, . . . , An), Tμ1k,...,μnk f (A1, . . . , An) → Tμ1,...,μn f (A1, . . . , An) in the oper-ator norm on L(X) as k → ∞.

Note:

(a) The weak convergence above is meant in the probabilist’s sense (see [1],p. 229).

(b) We can alternatively describe the conclusion of Theorem 1.1 as follows:The sequence of operational calculi specified by the sequence of n-tuples{(μ1k, . . . , μnk) : k = 1, . . . , ∞} converges as k → ∞ to the operationalcalculus specified by the n-tuple (μ1, . . . , μn).

We finish this introduction by briefly outlining the essential definitions andsome basic facts of the approach to Feynman’s operational calculi initiated in[4, 5]. A discussion of the heuristic ideas behind these operational calculi canbe found in Chapter 14 of [10].

Let X be a Banach space and let A1, . . . , An be nonzero bounded linearoperators on X. Except for the numbers ‖A1‖, . . . , ‖An‖, which will serveas weights, we ignore for the present the nature of A1, . . . , An as oper-ators and introduce a commutative Banach algebra consisting of ‘analytic


functions’ f (A1, . . . , An), where A1, . . . , An are treated as purely formalcommuting objects.

Consider the collection D = D(A1, . . . , An) of all expressions of the form

f (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mn Am11 · · · Amn

n (1.1)

where cm1,...,mn ∈ C for all m1, . . . , mn = 0, 1, . . . , and

‖ f (A1, . . . , An)‖ = ‖ f (A1, . . . , An)‖D(A1,...,An),

:=∞∑

m1,...,mn=0

|cm1,...,mn |‖A1‖m1 · · · ‖An‖mn < ∞. (1.2)

As pointed out in [4] the function on D(A1, . . . , An) defined by (1.2) makesD(A1, . . . , An) into a commutative Banach algebra under pointwise operations([4], Proposition 1.1). We refer to D(A1, . . . , An) as the disentangling algebraassociated with the n-tuple (A1, . . . , An) of bounded linear operators actingon X. This commutative Banach algebra will provide us with a frameworkwhere we can apply Feynman’s ‘rule’ (b) above rigorously rather than justheuristically.

Let μ1, . . . , μn be continuous probability measures defined at least onB([0, 1]), the Borel class of [0, 1]. The idea is to replace the operatorsA1, . . . , An with the elements A1, . . . , An from D and then form the desiredfunction of A1, . . . , An. Still working in D, we time order the expression forthe function and then pass to L(X) simply by removing the tildes.

Given nonnegative integers m1, . . . , mn, we let m = m1 + · · · + mn and

Pm1,...,mn(z1, . . . , zn) = zm11 · · · zmn

n . (1.3)

We are now ready to define the disentangling map Tμ1,...,μn which will carryus from our commutative framework to the noncommutative setting of L(X).

For j = 1, . . . , n and all s ∈ [0, 1], we take A j(s) = A j (recall that each A j isindependent of s) and, for i = 1, . . . , m, we define

Ci(s) :=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

A1(s) if i ∈ {1, . . . , m1},A2(s) if i ∈ {m1 + 1, . . . , m1 + m2},

......

An(s) if i ∈ {m1 + · · · + mn−1 + 1, . . . , m}.(1.4)

For each m = 0, 1, . . . , let Sm denote the set of all permutations of the integers{1, . . . , m}, and given π ∈ Sm, we let

�m(π) = {(s1, . . . , sm) ∈ [0, 1]m : 0 < sπ(1) < · · · < sπ(m) < 1}.Finally, we remark that we will use the notation μk to denote μ × · · · × μ︸︷︷︸

k times

.


Definition 1.2 Tμ1,...,μn

(Pm1,...,mn(A1, . . . , An)

)

:=∑

π∈Sm

∫

�m(π)

Cπ(m)(sπ(m)) · · · Cπ(1)(sπ(1))(μm11 × · · · × μmn

n )(ds1, . . . , dsm).

(1.5)

Then, for f (A1, . . . , An) ∈ D(A1, . . . , An) given by

f (A1, . . . , An) =∞∑

m1,...,mn=0

cm1,...,mn Am11 · · · Amn

n , (1.6)

we set Tμ1,...,μn

(f (A1, . . . , An)

)equal to

∞∑

m1,...,mn=0

cm1,...,mnTμ1,...,μn

(Pm1,...,mn(A1, . . . , An)

). (1.7)

Remark 1.3 Even though the A j’s are independent of s, the order of theoperator products in each term of (1.5) depends on the s’s and on the measuresμ1, . . . , μn. (If the A j’s do depend on s, we obtain exactly the same expressionas seen in (1.5) and then we have a nontrivial integrand. But this situation willconcern us only marginally in this paper. For details of the time dependentsetting see, for example, the papers [8, 15, 16].)

It is worth noting that the disentangling map as defined above is a linearoperator of norm one from D(A1, . . . , An) to L(X). (See [4].) In the commu-tative setting, the right-hand side of (1.5) gives us Am1

1 · · · Amnn , the expected

result of the commutative functional calculus [4, Proposition 2.2]. (Of course,commutativity allows us to write the m operators in any desired order.) Asis usual, we shall write the operator Tμ1,...,μn f in place of Tμ1,...,μn( f ) for anelement f of D(A1, . . . , An).

We shall sometimes write the bounded linear operator

Tμ1,...,μn

(f (A1, . . . , An)

)

as fμ1,...,μn(A1, . . . , An), fμ1,...,μn(A) with A denoting the n-tuple (A1, . . . , An)

of operators, or fμ(A) with μ denoting the n–tuple (μ1, . . . , μn) of measures.In particular,

Pm1...mnμ1,...,μn

(A) = Tμ1,...,μn

(Pm1,...,mn(A1, . . . , An)

). (1.8)

We find it convenient to use i as an index, so i denotes√−1. The real part of a

complex number z is written as �z and the imaginary part as �z. For a complexvector ζ = (ζ1, . . . , ζn) ∈ C

n, we set

�ζ = (�ζ1, . . . , �ζn), �ζ = (�ζ1, . . . , �ζn), |ζ | =√

|ζ1|2 + · · · + |ζn|2.

Remark 1.4 A family of Trotter product type formulas suitable for Feynman’soperational calculi (and mentioned in the abstract) will be established in


Theorem 2.4. Here the bounded linear operators A1, . . . , An need not be self-adjoint and X can be a Banach space. However, we will require the probabilitymeasures μ1, . . . , μn to be absolutely continuous with respect to Lebesguemeasure λ.

2 The Main Theorem

We give a detailed proof that for any n-tuple A = (A1, . . . , An) of self-adjointoperators, there exists r > 0 such that A is of ‘Paley–Wiener type’ (0, r, μ) forany μ = (μ1, . . . , μn). We will follow the statement of what this means with abrief discussion of its consequences for the enlargement of the domain of theassociated operational calculi.

Our main interest in this paper is in the Hilbert space setting. However, westate the definition of ‘Paley–Wiener type’ in the more general Banach spacesetting.

Definition 2.1 Let A1, . . . , An be bounded linear operators acting on a Banachspace X. Let μ = (μ1, . . . , μn) be an n-tuple of continuous probability mea-sures on B([0, 1]) and let

Tμ1,...,μn : D(A1, . . . , An) → L(X) (2.1)

be the disentangling map defined in Definition 1.2. If there exists C, r, s ≥ 0such that

‖Tμ1,...,μn

(ei(ζ,A)

)‖L(X) ≤ C(1 + |ζ |)ser|�ζ |, for all ζ ∈ C

n, (2.2)

then the n-tuple A = (A1, . . . , An) of operators is said to be of Paley–Wienertype (s, r, μ). (Note: Given A and ζ ∈ C

n, (ζ, A) = ζ1 A1 + · · · + ζn An.)

If the estimate (2.2) holds, then there exists a unique L(X)-valued distribu-tion Fμ,A ∈ L(C∞(Rn),L(X)) such that

Fμ,A( f ) = (2π)−n∫

RnTμ1,...,μn

(ei(ξ,A)

)f (ξ) dξ, (2.3)

for every rapidly decreasing function f ∈ S(Rn). Here

f (ξ) =∫

Rne−i(x,ξ) f (x) dx

denotes the Fourier transform of f . Moreover,

Fμ,A(Pm1,...,mn) = Pm1,...,mnμ1,...,μn

(A1, . . . , An), (2.4)

for all nonnegative integers m1, . . . , mn. Hence we have a rich extension ofthe functional calculus f �−→ fμ(A) from analytic functions with a uniformlyconvergent power series in a polydisk, to functions C∞ in a neighbourhood of


the support γμ(A) of Fμ,A. In fact, all of the distributions just mentioned arecompactly supported and so of finite order k. In the setting of Theorem 2.2,k will be the smallest integer strictly greater than n/2. (For example, k = 2,if n = 2 or 3.) Now let K = ∏n

j=1[−‖A j‖, ‖A j‖]. The functional calculus thenextends to all functions f that are k times continuously differentiable on someopen set containing K.

The support of γμ(A) of the distribution Fμ,A is defined as the μ–jointspectrum of the n–tuple A = (A1, . . . , An). The distribution Fμ,A is calledFeynman’s μ–functional calculus for A. The number rμ(A) = sup{|x| : x ∈γμ(A)} is called the μ–joint spectral radius of A. It is shown in [7] via Cliffordanalysis that the nonempty compact subset γμ(A) of R

n may be interpreted asthe set of singularities of a multidimensional analogue of the resolvent familyof a single operator. For more detail on or related to the last two paragraphs,see pages 186–192 and especially Theorem 3.1 and Proposition 3.2 of [6].

If we have more information about the particular operators and measuresthat are involved, we can sometimes further enlarge the functional calculus.Example 2.1, p.176–178 in [6] is an extreme case. Here A1 and A2 are the 2by 2, self-adjoint Pauli matrices σ1 and σ3. The measures μ1 and μ2 are anycontinuous probability measures with the support of μ1 entirely to the left ofthe support of μ2. In this case, fμ1,μ2(A1, A2) makes sense for any f which isdefined on the 4 point set {−1, 1} × {−1, 1}. This last set is the product of theordinary spectrums of σ1 and σ3.

Theorem 2.2 An n-tuple A = (A1, . . . , An) of bounded self-adjoint operatorsacting on a Hilbert space H is of Paley–Wiener type (0, r, μ) with r = (‖A1‖2 +· · · + ‖An‖2)1/2, for any n-tuple μ = (μ1, . . . , μn) of continuous probabilitymeasures on B([0, 1]).

Proof One of the keys to the proof is a use of the Martingale ConvergenceTheorem. We can apply it to the Radon–Nikodyn derivative of probabilitymeasures on [0, 1] which are absolutely continuous with respect to Lebesguemeasure λ. Such Radon–Nikodyn derivatives are nonnegative functions withL1(λ)-norm 1. However, we are only assuming that μ1, . . . , μn are continuousand so, given μi, we will begin by finding a sequence of absolutely continuousprobability measures which converge weakly to μi. (We will, but do not needto, do this even for the μi’s that are absolutely continuous with respect to λ.)

Let μ be a continuous Borel probability measure on [0, 1]. Let ρ : [0, 1] →R be a nonnegative continuous function with compact support in [0, 1) and‖ρ‖1 = 1. It will be convenient to let ρ(x) = 0 for x < 0. For every ε ∈ (0, 1],we set ρε(x) = ε−1ρ(x/ε), 0 ≤ x ≤ 1. It is easy to check that ‖ρε‖1 = 1 for 0 <

ε ≤ 1. Now we let

(ρε ∗ μ)(x) :=∫ x

0ρε(x − y)μ(dy), 0 ≤ x ≤ 1. (2.5)


We assert that (ρε ∗ μ)λ ⇀ μ as ε → 0+. Indeed, using the definition ofweak convergence, our assertion follows once we know that, for every con-tinuous function φ : [0, 1] → R,

∫ 1

0(ρε ∗ μ)(x)φ(x)dx →

∫ 1

0φ(x)μ(dx) as ε → 0+. (2.6)

We omit the proof of the limit (2.6) as it can be carried out using standard tech-niques. The basic idea of the proof is much like arguments using approximateidentities although some of the particular details follow Exercise 10, p. 194 of[17] more closely.

Now consider the partition of the interval [0, 1) into nk disjoint intervalsIk, = [( − 1)n−k, n−k), = 1, . . . , nk each of length n−k. (Below, n will bethe number of operator-measure pairs in our problem.) The collection Ak offinite disjoint unions of these intervals is an algebra – in fact a σ -algebra. Notethat Ak ⊂ Ak+1 for k = 1, 2, . . . . Let Pk : L1(λ) → L1(λ) be the conditionalexpectation operator [1, p.265] with respect to Ak; that is,

Pk f = nknk∑

=1

χIk,

∫

Ik,

f dλ. (2.7)

Note that the sequence {Pk f } is adapted with respect to the sequence {Ak}of σ -algebras [1, p. 280]. Then for each f ∈ L1(λ), by the Martingale Con-vergence Theorem [1, p. 285–286], Pk f → f in L1(λ) (and λ-a.e.) as k → ∞.

Further, by Theorem 10.1.3 from [1], we have∫ 1

0Pk f dλ =

∫ 1

0f dλ, k = 1, 2, . . . . (2.8)

(Remark The usual notation for Pk f in the probability literature is E( f |Ak).)Now set fi, j,k := Pk(ρ1/j ∗ μi) for each i = 1, . . . , n and j, k = 1, 2, . . . . Each

such step function fi, j,k is constant on each interval Ik, , = 1, . . . , nk and

fi, j,k := nknk∑

=1

χIk,

∫

Ik,

ρ1/j ∗ μi dλ. (2.9)

The function fi, j,k that we are about to define is a key to the proof:

fi, j,k := nknk−1−1∑

m=0

χIk,mn+i

∫

Ik−1,m+1

ρ1/j ∗ μi dλ. (2.10)

Note that this function has support in the finite union

Ji,k :=nk−1−1⋃

m=0

Ik,mn+i

of disjoint intervals, for each i = 1, . . . , n and Ji,k ∩ J ,k = ∅ for i �= . Theintegral

∫Ik−1,m+1

ρ1/j ∗ μi dλ in the sum (2.10) is a weight factor compensating


for the omission of terms from the sum (2.9), so that the following equalitieshold true:

∫ 1

0fi, j,k dλ =

∫ 1

0ρ1/j ∗ μi dλ =

∫ 1

0fi, j,k dλ. (2.11)

The first equality follows from (2.7), (2.8) and the definition of fi, j,k above.We turn now to the second equality in (2.11):

∫ 1

0fi, j,k dλ = nk

nk−1−1∑

m=0

1

nk

∫

Ik−1,m+1

ρ1/j ∗ μi dλ

=∫

Ik−1,1

ρ1/j ∗ μi dλ +∫

Ik−1,2

ρ1/j ∗ μi dλ + . . .

+∫

Ik−1,nk−1

ρ1/j ∗ μk dλ

=∫ 1

0ρ1/j ∗ μi dλ. (2.12)

Thus (2.11) is established.Now let φ be a continuous function on [0, 1]. Given ε > 0, use the uniform

continuity of φ to choose k so large that |φ(x) − φ(y)| < ε whenever |x − y| <

n−k+1. We wish to compare the integrals∫ 1

0 fi, j,kφ dλ and∫ 1

0 fi, j,kφ dλ. Webegin with the first of these. The RHS of the 2nd equality in (2.13) below isjust another way of writing the sum of the nk terms that appear on the LHS.

∫ 1

0fi, j,kφ dλ =

∫ 1

0

⎧⎨

⎩nknk∑

p=1

χIk,pφ

∫

Ik,p

ρ1/j ∗ μi dλ

⎫⎬

⎭dλ

=∫ 1

0nk

⎧⎨

⎩

nk−1−1∑

m=0

n∑

=1

χIk,mn+ φ

(∫

Ik,mn+

ρ1/j ∗ μi dλ

)⎫⎬

⎭ dλ

= nknk−1−1∑

m=0

n∑

=1

(∫

Ik,mn+

ρ1/j ∗ μi dλ

)(∫

Ik,mn+

φ dλ

). (2.13)

On the other hand, starting with (2.10) we have

∫ 1

0fi, j,kφ dλ =

∫ 1

0nk

nk−1−1∑

m=0

χIk,mn+iφ

(∫

Ik−1,m+1

ρ1/j ∗ μi dλ

)dλ

= nknk−1−1∑

m=0

(∫

Ik−1,m+1

ρ1/j ∗ μi dλ

)(∫

Ik,mn+i

φdλ

)

= nknk−1−1∑

m=0

n∑

=1

(∫

Ik,mn+

ρ1/j ∗ μi dλ

)(∫

Ik,mn+i

φ dλ

). (2.14)


First using (2.13) and (2.14) and then the Mean Value Theorem for Inte-grals, there exist ξk, ∈ Ik, , = 1, . . . , nk such that

∣∣∣∣∫ 1

0

(fi, j,k − fi, j,k

)φ dλ

∣∣∣∣

≤nk−1−1∑

m=0

n∑

=1

(∫

Ik,mn+

ρ1/j ∗ μi dλ

) ∣∣∣∣nk∫

Ik,mn+

φ dλ − nk∫

Ik,mn+i

φ dλ

∣∣∣∣

=nk−1−1∑

m=0

n∑

=1

∫

Ik,mn+

ρ1/j ∗ μi dλ∣∣φ(ξk,mn+

)− φ(ξk,mn+i

)∣∣ < ε. (2.15)

The final inequality follows from the fact that |ξk,mn+ − ξk,mn+i| < n−k+1 foreach i ∈ {1, . . . , n} and each j ∈ {1, 2, 3, . . . }. Thus for each i ∈ {1, . . . , n} andeach j ∈ {1, . . . , m},

fi, j,kλ − fi, j,kλ ⇀ 0 as k → ∞. (2.16)

Let μi, j,k := fi, j,kλ, i = 1, . . . , n and j, k = 1, 2, . . . .

Next we summarize what we have proved so far about the weak limits andthen draw some conclusions. For each i = 1, . . . , n, the following limits obtain.

A(ρ1/j ∗ μi

)λ ⇀ μi as j → ∞.

B fi, j,kλ = Pk(ρ1/j ∗ μi

)λ ⇀

(ρ1/j ∗ μi

)λ as k → ∞.

In fact, the probability densities in B converge in L1-norm and so themeasures converge in total variation norm and so certainly converge weakly.

C fi, j,kλ − fi, j,kλ ⇀ 0 as k → ∞.

From B and C we see that

D fi, j,kλ = Pk(ρ1/j ∗ μi

)λ ⇀

(ρ1/j ∗ μi

)λ as k → ∞.

Now from D and A we have the iterated weak limits,

E lim j→∞[limk→∞

(fi, j,k

)λ]

= lim j→∞ ρ1/j ∗ μi = μi.

Since [0, 1] is a separable metric space, it follows from E that there exists asequence ( fi, j,k( j ))λ such that for each i = 1, . . . , n,

F(

fi, j,k( j )

)λ ⇀ μi as j → ∞.

(See [1, p. 309–310] and especially the paragraph preceding Theorem 11.3.3.)Note that the increasing sequence k( j ), j = 1, 2, . . . , may be chosen indepen-dently of i = 1, . . . , n by observing that the space M([0, 1], R

n) of Rn-valued

Borel measures on [0, 1] is in duality with the space C([0, 1], Rn) of R

n-valuedcontinuous functions and that closed balls in M([0, 1], R

n) are compact andmetrizable for the associated weak*-topology.


Note As we continue we will just write j, k but will understand that k = k( j )as in F above.

The measure μi, j,k is given by

μi, j,k = fi, j,kλ = nknk−1−1∑

m=0

(χIk,mn+iλ

) ∫

Ik−1,m+1

ρ1/j ∗ μi dλ. (2.17)

Before beginning the calculation below we make some simple commentsand introduce some notation.

(a) Since exponential functions are entire, the exponential functions involvedbelow are certainly in the domain of the disentangling map.

(b) We will write (ζ, A) = ζ1 A1 + · · · + ζn An where ζ = (ζ1, . . . , ζn) is an n-tuple of complex numbers and A = (A1, . . . , An). Similar notation willbe used in connection with A.

(c) Since the disentangling algebra is commutative, we have

e(ζ,A) = eζ1 A1+···+ζn An = eζ1 A1 . . . eζn An .

Because of how the function fi, j,k is supported (see (2.10)) and since μi, j,k isdefined using fi, j,k, an extension of Proposition 2.2 from [6] allows us to do thefollowing calculation (much as was done in Example 2.2 of that paper):

Tμ1, j,k,...,μn, j,k(ei(ζ,A))

=(

exp

{iζn

(∫

Ik−1,nk−1

ρ1/j ∗ μndλ

)An

}. . .

exp

{iζ1

(∫

Ik−1,nk−1

ρ1/j ∗ μ1dλ

)A1

}). . .

(exp

{iζn

(∫

Ik−1,2

ρ1/j ∗ μndλ

)An

}. . .

exp

{iζ1

(∫

Ik−1,2

ρ1/j ∗ μ1dλ

)A1

}).

(exp

{iζn

(∫

Ik−1,1

ρ1/j ∗ μndλ

)An

}. . .

exp

{iζ1

(∫

Ik−1,1

ρ1/j ∗ μ1dλ

)A1

}). (2.18)

Breaking ζ1 = ξ1 + iη1 into real and imaginary parts, we have iζ1 = −η1+iξ1. Further, |eiζ1 | = |eiξ1 ||e−η1 | ≤ e|η1| = e|�ζ1|. Similarly, |eiζ2 | ≤ e|�ζ2|, . . . ,|eiζn | ≤ e|�ζn|.


Now using (2.18), the self-adjointness of Ai, i = 1, . . . , n, the standardmultiplicative Banach algebra inequality and the simple computations justabove, we can write

‖Tμ1, j,k,...,μn, j,k(ei(ζ,A))‖

≤∥∥∥∥∥exp

{iζn

(∫

Ik−1,nk−1

ρ1/j ∗ μndλ

)An

}∥∥∥∥∥ . . .

∥∥∥∥∥exp

{iζ1

(∫

Ik−1,nk−1

ρ1/j ∗ μ1dλ

)A1

}∥∥∥∥∥ . . .

∥∥∥∥exp

{iζn

(∫

Ik−1,2

ρ1/j ∗ μndλ

)An

}∥∥∥∥ . . .

∥∥∥∥exp

{iζ1

(∫

Ik−1,2

ρ1/j ∗ μ1dλ

)A1

}∥∥∥∥ .

∥∥∥∥exp

{iζn

(∫

Ik−1,1

ρ1/j ∗ μndλ

)An

}∥∥∥∥ . . .

∥∥∥∥exp

{iζ1

(∫

Ik−1,1

ρ1/j ∗ μ1dλ

)A1

}∥∥∥∥ (2.19)

≤ e|�ζn|

(∫Ik−1,nk−1

ρ1/j∗μndλ

)‖An‖

. . . e|�ζ1|

(∫Ik−1,nk−1

ρ1/j∗μ1dλ

)‖A1‖

. . .

e|�ζn|

(∫Ik−1,2

ρ1/j∗μndλ)‖An‖

. . . e|�ζ1|

(∫Ik−1,2

ρ1/j∗μ1dλ)‖A1‖

.

e|�ζn|

(∫Ik−1,1

ρ1/j∗μndλ)‖An‖

. . . e|�ζ1|

(∫Ik−1,1

ρ1/j∗μ1dλ)‖A1‖

.

Now we are dealing with numbers and so the noncommutativity of theoperators is not an issue. Since

∫

Ik−1,1

ρ1/j ∗ μ1dλ +∫

Ik−1,2

ρ1/j ∗ μ1dλ + . . .

+∫

Ik−1,nk−1

ρ1/j ∗ μ1dλ =∫ 1

0ρ1/j ∗ μ1dλ = 1,

the product of the terms in the column furthest to the right is e|�ζ1|‖A1‖.A similar thing happens in the other columns and so we obtain

‖Tμ1, j,k,...,μn, j,k(ei(ζ,A))‖ ≤ e(|�ζ1|‖A1‖+|�ζ2‖A2‖+···+|�ζn|‖An‖)

≤ e[|�ζ1|2+···+|�ζn|2]1/2[‖A1‖2+···+‖An‖2]1/2 = er|�ζ | (2.20)


where r = [‖A1‖2 + · · · + ‖An‖2]1/2 and |�ζ | is the Euclidean norm in Rn of

the vector (�ζ1, . . . , �ζn).

Theorem 1.1 and the inequality (2.20) show that the bounded self-adjoint operators A1, . . . , An are of Paley–Wiener type (0, r, μ) where μ =(μ1, . . . , μn). ��

Remark 2.3 Following a suggestion of Jefferies, Johnson and Nielsen usedthe main theorem from [11], that is, Theorem 1.1 of this paper, to prove thespecial case of Theorem 2.2 where n = 2 and the operational calculus is theWeyl calculus. In the closing remark of [11], Johnson and Nielsen asserted thatan argument similar to the one in that paper would take care of the case ofthe Weyl calculus for any n. In fact, the argument in [11] fails at a critical pointfor n ≥ 3.

The following result is a simple consequence of Theorem 1.1 and the proofof Theorem 2.2.

Corollary 2.1 Let A1, . . . , An be bounded, self-adjoint operators on the Hilbertspace H and let μ1, . . . , μn be absolutely continuous probability measures onB([0, 1]). Finally let ξ = (ξ1, . . . ξn) be an n-tuple of real numbers. Then

‖Tμ1,...,μn(ei(ξ,A))‖ = ‖Tμ1,...μn(e

iξ1 A1 · · · eiξn An)‖ = 1. (2.21)

Proof Let ζ from Theorem 2.2 (see also Definition 2.1) equal ξ as above. Thenwe see that the operator on the RHS of (2.18) is unitary and so has norm 1.But, by Theorem 1.1 and the fact that μi, j,k ⇀ μi for each i = 1, . . . , n, wehave the operator norm convergence of the sequence of operators in (2.18)to Tμ1,...,μn(e

i(ξ,A)). The equality in (2.21) follows. ��

The reader may have noticed that we did not need the self-adjointness of theoperators nor even the Hilbert space setting until late in the proof of Theorem2.2. In fact, we can use some of the early parts of the proof to establish a classof “Trotter product formulas” suitable for Feynman’s operational calculi in thegeneral Banach space setting.

Theorem 2.4 Let X be a Banach space over C and let ζ = (ζ1, . . . , ζn) bean n-tuple of complex numbers. Further, let μ1, . . . , μn be probability mea-sures on B([0, 1]) each of which is absolutely continuous with respect to λ;hence, μ1 = g1λ, . . . , μn = gnλ where g1, . . . , gn are nonnegative functions inL1([0, 1],B([0, 1]), λ). Finally, let A = (A1, . . . , An) be an n-tuple of boundedlinear operators on X. Then

‖Tμ1,...,μn(ei(ζ,A)) − Tμ1,k,...,μn,k(e

i(ζ,A))‖L(X) → 0 (2.22)


as k → ∞ where

Tμ1,k,...,μn,k(ei(ζ,A)) = Tμ1,k,...,μn,k(e

iζ1 A1 . . . eiζn An)

= exp

{iζn

(∫

Ik−1,nk−1

gndλ

)An

}. . . exp

{iζ1

(∫

Ik−1,nk−1g1dλ

)A1

}. . .

exp

{iζn

(∫

Ik−1,2

gndλ

)An

}. . . exp

{iζ1

(∫

Ik−1,2

g1dλ

)A1

}

exp

{iζn

(∫

Ik−1,1

gndλ

)An

}. . . exp

{iζ1

(∫

Ik−1,1

g1dλ

)A1

}. (2.23)

Proof We will just comment on which parts of the proof of Theorem 2.2 areneeded here and which are not. We will also note the point at which Theorem1.1 is applied.

The first part of the proof of Theorem 2.2 is needed only for measures whichare continuous but not absolutely continuous. We have no such measures here.The second part of the earlier proof involves the Martingale ConvergenceTheorem. We need this and we obtain fi,k = Pkgi, i = 1, . . . , n with (2.9)suitably adjusted. (Note that we do not need the index j as our measures areall absolutely continuous.) The function fi,k is then defined with the integrandsin (2.10) changed to gi. The proof now moves along with no j’s involved andwith ρ1/j ∗ μi replaced by gi, i = 1, . . . , n.

When we reach the summary A-F, A is not needed. Replace B with fi,kλ =(Pkgi)λ ⇀ giλ as k → ∞ for i = 1, . . . , n. Change C to fi,kλ − fi,kλ ⇀ 0 ask → ∞ for i = 1, . . . , n. Based on the revised form of B and C, change D tofi,kλ ⇀ giλ as k → ∞; that is

μi,k ⇀ μi

as k → ∞ for i = 1, . . . , n. (Note that in (2.17) μi, j,k, fi, j,k and ρ1/j ∗ μi arereplaced by μi,k, fi,k and gi, respectively.)

The calculation in (2.18) is done in the same way but the subscript j on theLHS is missing and ρ1/j ∗ μi is replaced by gi, i = 1, . . . , n, on the RHS.

We can now apply Theorem 1.1 to finish the proof of this theorem. ��

Remark 2.5

(a) The inequalities which concerned us toward the end of the proof ofTheorem 2.2 did not concern us in the proof of Theorem 2.4 sincewe were not trying to show that (A1, . . . , An) is of Paley–Wiener type(0, r, (μ1, . . . , μn)).

(b) The Trotter products on the RHS of (2.23) look more like the usualTrotter products when special choices are made for ζ . Some examples:(1) Each ζ j equals i (or ti). (2) Each ζ j equals −1 (or −t).

(c) We hope to investigate variations and consequences of Theorem 2.4 inlater work.


Remark 2.6 In view of Corollary 3.1 of the paper [14], we may write (2.22) as

‖Tλ,...,λ(ei(ζ,[g·A])) − Tμ1,k,...,μn,k(ei(ζ,A))‖L(X) → 0 (2.24)

where [g · A] :=(

˜[g1 · A1], . . . , ˜[gn · An])

, that is the time independent opera-tors A1, . . . , An are replaced by the time dependent operators g1 · A1, . . . , gn ·An where g1 = dμ1

dλ, . . . , gn = dμn

dλ.

We now present two simple examples illustrating Theorem 2.4.

Example 2.7 For the first example we assume that, in Theorem 2.4, g1 = · · · =gn = 1; i.e. Lebesgue measure is associated to each operator. It follows from[5, Lemma 5.4] that

Tμ1,...,μn(ei(ζ,A)) = Tλ,...,λ(ei(ζ,A)) = ei(ζ,A) (2.25)

and so Theorem 2.4 tells us that

ei(ζ,A) = limk→∞

{exp

{ζn

nk−1An

}. . . exp

{ζ1

nk−1A1

}}nk−1

(2.26)

Example 2.8 In the second example, we will consider two operators, A1 andA2 and we will take μ1 = 2t dλ and μ2 = 3t2 dλ. For any nonnegative integer l,we have

∫

Ik−1,l

g1 dλ = 2l − 1

22k−2, (2.27)

and∫

Ik−1,l

g2 dλ = 3l2 − 3l + 1

23k−3. (2.28)

Theorem 2.4 tells us that

limk→∞

Tμ1,k,μ2,k ei(ζ1 A1+ζ2 A2)

= limk→∞

{exp

(iζ2

[3 · 22k−2 − 3 · 2k−1 + 1

23k−3

]A2

)·

exp

(iζ1

[2 · 2k−1 − 1

22k−2

]A1

)· · ·

exp

(iζ2

[7

23k−3

]A2

)exp

(iζ1

[3

22k−2

]A1

)·

exp

(iζ2

[1

23k−3

]A2

)exp

(iζ1

[1

22k−2

]A1

)}

= Tμ1,μ2 ei(ζ1 A1+ζ2 A2). (2.29)


Using Corollary 3.1 of [14], the last expression immediately above can be

written as Tλ,λei(ζ1 ˜[g1·A1]+ζ2 ˜[g2·A2]) where the time dependence is carried by thefunctions g1 and g2.

It is clear from Example 2.8 that we can produce an infinite number ofdistinct Trotter product formulas by varying the probability densities g1 and g2.

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feynman’s operational calculi: spectral theory for noncommuting self-adjoint operators

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